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Math Help - Why Root Pi?!?! (In non-integer factorials)

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    Why Root Pi?!?! (In non-integer factorials)

    Consider the problem 3.5! : I can understand the concept of having:

    3.5!=(7/2)*(5/2)*(3/2)*(1/2)

    but in the real solution, it is actually:



    Now can someone tell me why is the root of pi used and how did it originate in the use of finding non-integer factorials?
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  2. #2
    TD!
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    Quote Originally Posted by anthmoo View Post
    Consider the problem 3.5! : I can understand the concept of having:

    3.5!=(7/2)*(5/2)*(3/2)*(1/2)
    This isn't correct, the definition n! = n*(n-1)! only holds if n is a natural number, this is the factorial.
    It can be generalized to the "Gamma-function", defined for all real numbers except negative integers.

    Your examaple (Gamma(1/2) = sqrt(pi)) can be elegantly showed if you know the relation between the Gamma and the Beta function.
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    Quote Originally Posted by TD! View Post
    This isn't correct, the definition n! = n*(n-1)! only holds if n is a natural number, this is the factorial.
    It can be generalized to the "Gamma-function", defined for all real numbers except negative integers.

    Your examaple (Gamma(1/2) = sqrt(pi)) can be elegantly showed if you know the relation between the Gamma and the Beta function.
    I know that isn't correct it is just the use of pi that confuses me.

    Why is it there? My main understanding of pi comes from the fact that it is how many times a circles diameter fits in it's circumference and I can't possibly see how the value of pi applies to this situation!
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    TD!
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    You'll encounter pi in the (seemingly) most strange places in mathematics, without directly seeing a link to circles.
    For example, if you add 1+1/4+1/9+1/16+1/25+...+1/n²+... (so the inverse squares, "all" of them), you get pi²/6
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    And there is no explanation for it?!

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    TD!
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    What do you want as an explanation?

    I can give you the definition of the Gamma-function and from that definition it will follow that if you take Gamma(1/2), you get sqrt(pi).
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    I just want to know if there is an explanation for Pi cropping up everywhere

    Sorry if I don't seem grateful - I am very grateful! That "1+1/4+1/9+1/16+1/25+...+1/nē+..." you gave me is driving me insane and I fear that Pi may be my obsession for life now!
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    TD!
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    I don't know your level of mathematics, but to compute these things (such as Gamma(1/2 or 1+1/4+1/9+...), you need some knowledge of calculus.

    If you ever start doing complex analysis, you'll see pi dropping by in almost all important formula's.
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    Quote Originally Posted by anthmoo View Post
    I just want to know if there is an explanation for Pi cropping up everywhere
    \pi happens to be a very interesting number because it does appear in many strange cases all over mathematics. Usually infinite sums involve this number quite a lot.

    The reason why .5! involves pi is because it is the way the extended definition of the factorial was defined. I never heard of anybody else being succesful in using a different definition for the extended factorial.
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